$$ \text { If } A \in R^{m \times n}, \text { then } A^{T} \in ? \text { . Find the transpose of the matrix } A=\left[\begin{array}{ll} 2 & 0 \\ 7 & 1 \\ 4 & 9 \end{array}\right] \text { . } $$

Understanding the dimensions of a matrix is fundamental to operations like transposition. In matrix terminology, dimensions are described by the number of rows and columns it possesses. For instance, if a matrix has 3 rows and 2 columns, we denote it as a \(3 \times 2\) matrix.

Let's consider our given matrix \(A\):
$$A = \begin{pmatrix} 2 & 0 \ 7 & 1 \ 4 & 9 \end{pmatrix}$$
Here, it's crucial to note that this matrix has 3 rows and 2 columns, making it a \(3 \times 2\) matrix. Whenever we talk about dimensions, we always mention rows first and columns second.

Knowing the dimensions helps in understanding how different matrix operations will affect any given matrix. Especially when you are performing operations like addition or multiplication, it's important to ensure the dimensions are compatible.

A transpose of a matrix is a fundamental concept in linear algebra. To transpose a matrix means to flip it over its diagonal. This means that the row and column indices of each element are swapped. If you have a matrix \(A\) with dimensions \(m \times n\), its transpose (denoted as \(A^T\)) will have dimensions \(n \times m\).

Let's look at our matrix \(A\):
$$A = \begin{pmatrix} 2 & 0 \ 7 & 1 \ 4 & 9 \end{pmatrix}$$
To find \(A^T\), we swap its rows and columns. The first row [2, 0] becomes the first column of \(A^T\). The second row [7, 1] becomes the second column of \(A^T\). The third row [4, 9] becomes the third column of \(A^T\). So the transpose matrix \(A^T\) will be:
$$A^T = \begin{pmatrix} 2 & 7 & 4 \ 0 & 1 & 9 \end{pmatrix}$$

As you can see, the dimensions also change from \(3 \times 2\) to \(2 \times 3\), as per the rule.

To understand matrix transpose deeply, the concept of row-column swap is key. When we take the transpose of a matrix, essentially, each element \(a_{ij}\) in the original matrix \(A\) ends up at position \(a_{ji}\) in the transposed matrix \(A^T\).

Let's break down our example:
- The element at the 1st row and 1st column in matrix \(A\) (which is 2) moves to the 1st row, 1st column position in \(A^T\).
- The element at the 1st row, 2nd column in matrix \(A\) (which is 0) moves to the 2nd row, 1st column in \(A^T\).
- The element at the 2nd row, 1st column (which is 7) moves to the 1st row, 2nd column in \(A^T\).
- Continuing this process for all elements gives us the transposed matrix:
$$ A^T = \begin{pmatrix} 2 & 7 & 4 \ 0 & 1 & 9 \end{pmatrix} $$
This row-column swap makes it incredibly easy to visualize how the transpose of any matrix will look. It simply flips the structure of the matrix over its diagonal.